Finite volume methods

Eymard, Robert; Gallouët, Thierry; Herbin, Raphaèle

doi:10.1016/s1570-8659(00)07005-8

Cited by 1,355 publications

(1,541 citation statements)

References 76 publications

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“…The corresponding (homogeneous) fluxes f 1 (x) and f 2 (x) can be readily determined from (2). Similar expressions hold for D = 0 or D < 0.…”

Section: Modification Of the Inhomogeneous Fluxmentioning

confidence: 87%

“…For space discretization of the conservation law we employ the finite volume method (FVM); see [2] for a detailed account. For the numerical fluxes in the discrete conservation law there exist many schemes.…”

Section: Introductionmentioning

confidence: 99%

“…The conservation law (1) can be concisely written as df /dx = cϕ + s(ϕ) with the flux f defined in (2). In the FVM we cover the domain with a finite number of control volumes (cells) I j of size ∆x.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complete Flux Scheme for Conservation Laws Containing a Linear Source

Boonkkamp

Kumar

et al. 2016

Lecture Notes in Computational Science and Engineering

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“…The corresponding (homogeneous) fluxes f 1 (x) and f 2 (x) can be readily determined from (2). Similar expressions hold for D = 0 or D < 0.…”

Section: Modification Of the Inhomogeneous Fluxmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complete Flux Scheme for Conservation Laws Containing a Linear Source

Boonkkamp

Kumar

et al. 2016

Lecture Notes in Computational Science and Engineering

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“…monotone flux function with respect to φ in the sense of the following definition (see [13] for the theory and some examples). 2. g is Lipschitz-continuous with respect to both variables over R,…”

Section: Discretization Of the Nonlinear Advection-diffusion Equationmentioning

confidence: 99%

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

Gastaldo

Herbin

Latché

2009

IMA Journal of Numerical Analysis

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“…Eymard et al [36], Aavatsmark et al [1], or Droniou and Eymard [32], mimetic finite difference; cf. Brezzi et al [21], covolume; cf.…”

Section: Introductionmentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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Finite volume methods

Cited by 1,355 publications

References 76 publications

Complete Flux Scheme for Conservation Laws Containing a Linear Source

Complete Flux Scheme for Conservation Laws Containing a Linear Source

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

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